Space Situational Awareness (SSA) includes knowledge of the near-Earth space environment which can be accomplished through the tracking and identification of Earth-orbiting space objects to protect space assets and maintain awareness of potentially adversarial space deployments. Fundamental to the success of the SSA mission is the rigorous inclusion of uncertainty in the space surveillance network (SSN). The proper characterization and robust quantification of uncertainty is a common requirement to many SSA functions including tracking and data association, resolution of uncorrelated tracks (UCTs), conjunction analysis and probability of collision, sensor resource management, and anomaly detection. Other fields of endevour in which estimating the state of an object and its uncertainty include but are not limited to aeronautics and aviation, air and missile defense, etc.
The general framework for the problem of estimating the state of an object and its uncertainty (i.e., its probability density function) in the presence of uncertainty in the dynamics, model parameters, and initial conditions is that of the general Bayesian nonlinear filter. The Kalman filter optimally characterizes the state and uncertainty in the linear Gaussian case and the extended Kalman filter approximately characterizes the state in the Gaussian and (mildly) nonlinear case. Higher order filters can also be employed to improve the characterization of the state. The uncertainty propagation component of these filters uses the Ricatti equation or a higher order Ricatti equation (e.g., the covariance matrix associated with the extended Kalman filter is obtained as a solution of the Ricatti equation).
A different class of methods starts from the representation of the initial probability density function by a set of particles or states, propagates the ensemble of particles through the differential equation defining the dynamics (i.e., the time evolution of the particles or states) over some period of time, and then recovers the characterization of uncertainty at the end time. Mathematically speaking, the propagation of a given state amounts to solving an initial value problem (IVP). The particles can be chosen randomly through sampling techniques as in Monte Carlo methods (e.g., particle filters) or deterministically as in the unscented transform of the unscented Kalman filter or higher order methods based on Gauss-Hermite quadrature. Gaussian sum filters and generalizations using mixtures can also utilize the unscented Kalman filter as the basis for propagating the individual probability density functions. Grid methods represent the uncertainty through the propagation of particles and the subsequent recovery of the uncertainty. Ultimately, either the Fokker-Planck equation (for continuous time dynamics) or Chapman-Kolmogorov equation (for discrete time dynamics) govern the evolution of the probability density function. In the former, this parabolic partial differential equation can be solved by the method of lines in which the spatial variables are discretized with the time variables remaining continuous, resulting in an ensemble of IVPs. Finally, in addition to the general nonlinear Bayesian filter, the Probability Hypothesis Density filter and its generalization, the Cardinalized Probability Hypothesis Density filter can make use of Monte Carlo methods with the corresponding propagation of an ensemble of particles. Thus, a large class of methods for propagating a state and its uncertainty require the propagation of an ensemble of particles or states through the nonlinear dynamics. Once recovered and characterized, the uncertainty can be used to update the nonlinear filter, to compute a likelihood function or ratio for data association, to compute a probability of collision in conjunction analysis, to assist with the detection of anomalies such as maneuvers, and to provide state uncertainty for sensor resource management.
Traditional algorithms for propagating an ensemble of states are based on explicit numerical methods. As such, they solve each IVP independently from one another, regardless of the proximity of the initial conditions, by calculating the state of a system at a later time from the state of the system at an earlier time. Hence, even if two initial conditions differed by an infinitesimal amount and the solution to the first IVP is known because it had already been computed, current approaches would not be able to exploit this information in order to solve the second IVP in a more computationally efficient manner. Since the IVPs that arise in uncertainty propagation usually contain initial conditions that are closely spaced about some estimate of the current state, approaches are needed that can reduce the computational cost of uncertainty propagation by exploiting such proximity.